Exercise 13. Consider an exchange economy with I consumers and L goods. Suppose that each consumeri has utility function ultra, ,ari) over RE and initial endowment wi >> 0. We assume that each u"(a.') is strictly increasing in each dimension. Given a feasible allocation :3: = (3:1, . - - ,a'l), say thati envysj in 1' ifua) :> vim"). An allocation .1: is called enwfree if there exists no pair of consumers i and j such thati envys j in 2:. 1. Is every competitive muilibrium allocation envyfree? Either prove that this is true or briey explain why this is not true. 2. An allocation 3: contains an envy cycle if there exists a set of consumers {i11 - - - ,ik} such that i1 : i,\" and ui"(a:i"+1) > uari\") in .v for each n : 1, ,h. Can a competitive equilibrium allocation in this economy contain an envy cycle? Either prove that this is impossible or briey explain why this is possible. 3. Consider a social planner whose objective is to ensure that all allocations are envy free. Suppose that the planner can intervene by changing consumers' initial endow ment, subject to the floral social endowment. Can the planner find an intervention such that, after the intervention, every competitive equilibrium application is envy free? Either prove that this is true, or briefly eaplain why this is not true. 4. Now let I = L = 2. Consumer 1's endowment is (D, 1] where 0 is consumer 1's endowment of goods I and 1 his endowment of goods 2. Consumer 2's endowment is (1,0). Consumer 1's utility function is u1(a.'},.v%) = ln(s:}) + 2111(I), where so} > D is consumer 1's consumption of goods 1 and 3:; :> 0 his consumption of goods 2. Consumer 2's utility function has the form u2(:o,a:) = ($)55(a:%)55. Find the competitive equilibrium in this economy (normalise the price of goods I by 1.)